openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /QuotientIntersection.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.LocalComparison | |
| noncomputable section | |
| variable {K R B : Type*} [Field K] [CommRing R] [CommRing B] | |
| [Algebra K R] [Algebra K B] | |
| def principalIntersectionEquiv (π : R →ₐ[K] B) (hπ : Function.Surjective π) | |
| (g : R) (hg : RingHom.ker π.toRingHom = Ideal.span {g}) (f : R) : | |
| (R ⧸ Ideal.span {f,g}) ≃ₐ[K] (B ⧸ Ideal.span {π f}) := by | |
| let q := (Ideal.Quotient.mkₐ K (Ideal.span {π f})).comp π | |
| have hk : RingHom.ker q.toRingHom = Ideal.span {f,g} := by | |
| have hq : RingHom.ker q.toRingHom = (Ideal.span {π f}).comap π.toRingHom := by | |
| ext x | |
| change Ideal.Quotient.mk (Ideal.span {π f}) (π x) = 0 ↔ _ | |
| exact Ideal.Quotient.eq_zero_iff_mem | |
| rw [hq] | |
| have he : (Ideal.span {f}).map π.toRingHom = Ideal.span {π f} := by | |
| rw [Ideal.map_span, Set.image_singleton] | |
| rfl | |
| rw [← he, Ideal.comap_map_of_surjective π.toRingHom hπ] | |
| change Ideal.span {f} ⊔ RingHom.ker π.toRingHom = _ | |
| rw [hg, ← Ideal.span_union] | |
| congr 1 | |
| exact (Ideal.quotientEquivAlgOfEq K hk.symm).trans | |
| (Ideal.quotientKerAlgEquivOfSurjective | |
| ((Ideal.Quotient.mk_surjective).comp hπ)) | |
| end | |
| end MaximalSeshadri.LocalComparison | |
| end | |
| end | |
| end | |
| end OAI | |